12. $$ \int_{1}^{e} x^{4} \ln x d x $$

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Calculate the integral $$ \int_{1}^{e} x^{4} \ln x dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
  $$\int_{1}^{e} x^{4}\ln{\left(xight)} dx$$
- step1: Evaluate the integral:
  $$\int x^{4}\ln{\left(xight)} dx$$
- step2: Prepare for integration by parts:
  $$\begin{align}&u=\ln{\left(xight)}\&dv=x^{4}dx\end{align}$$
- step3: Calculate the derivative:
  $$\begin{align}&du=\frac{1}{x}dx\&dv=x^{4}dx\end{align}$$
- step4: Evaluate the integral:
  $$\begin{align}&du=\frac{1}{x}dx\&v=\frac{x^{5}}{5}\end{align}$$
- step5: Substitute the values into formula:
  $$\ln{\left(xight)}	imes \frac{x^{5}}{5}-\int \frac{1}{x}	imes \frac{x^{5}}{5} dx$$
- step6: Calculate:
  $$\frac{x^{5}\ln{\left(xight)}}{5}-\int \frac{x^{4}}{5} dx$$
- step7: Evaluate the integral:
  $$\frac{x^{5}\ln{\left(xight)}}{5}-\frac{x^{5}}{25}$$
- step8: Simplify the expression:
  $$\frac{1}{5}x^{5}\ln{\left(xight)}-\frac{x^{5}}{25}$$
- step9: Return the limits:
  $$\left(\frac{1}{5}x^{5}\ln{\left(xight)}-\frac{x^{5}}{25}ight)\bigg |_{1}^{e}$$
- step10: Calculate the value:
  $$\frac{4e^{5}+1}{25}$$
The integral of $$x^{4} \ln x$$ from 1 to $$e$$ is $$\frac{4e^{5}+1}{25}$$.
The integral of $$ x^{4} \ln x $$ from 1 to $$ e $$ is $$ \frac{4e^{5}+1}{25} $$.

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